X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Freduction%2Fcpr.ma;h=3c0514a3c7a876b4ca2c0f1a305b54d837e3f621;hb=32bdf7f107be22a121fab8225c5fae4eb6b33633;hp=e6ad7b2bac33235f086ad727f68cad5ec9152609;hpb=09af7a9751464291ec3f32fb80c92fe1accdbf88;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpr.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpr.ma
index e6ad7b2ba..3c0514a3c 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpr.ma
@@ -12,49 +12,49 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/unfold/cpqs.ma".
+include "basic_2/notation/relations/pred_4.ma".
+include "basic_2/grammar/genv.ma".
+include "basic_2/grammar/cl_shift.ma".
+include "basic_2/relocation/ldrop_append.ma".
+include "basic_2/relocation/lsubr.ma".
 
 (* CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS ***************************)
 
-(* Basic_1: includes: pr0_delta1 pr2_delta1 pr2_thin_dx pr2_head_1 *)
+(* activate genv *)
+(* Basic_1: includes: pr0_delta1 pr2_delta1 pr2_thin_dx *)
 (* Note: cpr_flat: does not hold in basic_1 *)
-inductive cpr: lenv → relation term ≝
-| cpr_atom : ∀I,L. cpr L (⓪{I}) (⓪{I})
-| cpr_delta: ∀L,K,V,V2,W2,i.
-             ⇩[0, i] L ≡ K. ⓓV → cpr K V V2 →
-             ⇧[0, i + 1] V2 ≡ W2 → cpr L (#i) W2
-| cpr_bind : ∀a,I,L,V1,V2,T1,T2.
-             cpr L V1 V2 → cpr (L. ⓑ{I} V1) T1 T2 →
-             cpr L (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
-| cpr_flat : ∀I,L,V1,V2,T1,T2.
-             cpr L V1 V2 → cpr L T1 T2 →
-             cpr L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
-| cpr_zeta : ∀L,V,T1,T,T2. cpr (L.ⓓV) T1 T →
-             ⇧[0, 1] T2 ≡ T → cpr L (+ⓓV. T1) T2
-| cpr_tau  : ∀L,V,T1,T2. cpr L T1 T2 → cpr L (ⓝV. T1) T2
-| cpr_beta : ∀a,L,V1,V2,W,T1,T2.
-             cpr L V1 V2 → cpr (L.ⓛW) T1 T2 →
-             cpr L (ⓐV1. ⓛ{a}W. T1) (ⓓ{a}V2. T2)
-| cpr_theta: ∀a,L,V1,V,V2,W1,W2,T1,T2.
-             cpr L V1 V → ⇧[0, 1] V ≡ V2 → cpr L W1 W2 → cpr (L.ⓓW1) T1 T2 →
-             cpr L (ⓐV1. ⓓ{a}W1. T1) (ⓓ{a}W2. ⓐV2. T2)
+inductive cpr: relation4 genv lenv term term ≝
+| cpr_atom : ∀I,G,L. cpr G L (⓪{I}) (⓪{I})
+| cpr_delta: ∀G,L,K,V,V2,W2,i.
+             ⇩[0, i] L ≡ K. ⓓV → cpr G K V V2 →
+             ⇧[0, i + 1] V2 ≡ W2 → cpr G L (#i) W2
+| cpr_bind : ∀a,I,G,L,V1,V2,T1,T2.
+             cpr G L V1 V2 → cpr G (L.ⓑ{I}V1) T1 T2 →
+             cpr G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
+| cpr_flat : ∀I,G,L,V1,V2,T1,T2.
+             cpr G L V1 V2 → cpr G L T1 T2 →
+             cpr G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+| cpr_zeta : ∀G,L,V,T1,T,T2. cpr G (L.ⓓV) T1 T →
+             ⇧[0, 1] T2 ≡ T → cpr G L (+ⓓV.T1) T2
+| cpr_tau  : ∀G,L,V,T1,T2. cpr G L T1 T2 → cpr G L (ⓝV.T1) T2
+| cpr_beta : ∀a,G,L,V1,V2,W1,W2,T1,T2.
+             cpr G L V1 V2 → cpr G L W1 W2 → cpr G (L.ⓛW1) T1 T2 →
+             cpr G L (ⓐV1.ⓛ{a}W1.T1) (ⓓ{a}ⓝW2.V2.T2)
+| cpr_theta: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
+             cpr G L V1 V → ⇧[0, 1] V ≡ V2 → cpr G L W1 W2 → cpr G (L.ⓓW1) T1 T2 →
+             cpr G L (ⓐV1.ⓓ{a}W1.T1) (ⓓ{a}W2.ⓐV2.T2)
 .
 
 interpretation "context-sensitive parallel reduction (term)"
-   'PRed L T1 T2 = (cpr L T1 T2).
+   'PRed G L T1 T2 = (cpr G L T1 T2).
 
 (* Basic properties *********************************************************)
 
-(* Note: it does not hold replacing |L1| with |L2| *)
-lemma cpr_lsubr_trans: ∀L1,T1,T2. L1 ⊢ T1 ➡ T2 →
-                       ∀L2. L2 ⊑ [0, |L1|] L1 → L2 ⊢ T1 ➡ T2.
-#L1 #T1 #T2 #H elim H -L1 -T1 -T2
+lemma lsubr_cpr_trans: ∀G. lsub_trans … (cpr G) lsubr.
+#G #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2
 [ //
-| #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
-  lapply (ldrop_fwd_ldrop2_length … HLK1) #Hi
-  lapply (ldrop_fwd_O1_length … HLK1) #H2i
-  elim (ldrop_lsubr_ldrop2_abbr … HL12 … HLK1 ? ?) -HL12 -HLK1 // -Hi
-  <H2i -H2i <minus_plus_m_m /3 width=6/
+| #G #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
+  elim (lsubr_fwd_ldrop2_abbr … HL12 … HLK1) -L1 * /3 width=6/
 |3,7: /4 width=1/
 |4,6: /3 width=1/
 |5,8: /4 width=3/
@@ -62,205 +62,203 @@ lemma cpr_lsubr_trans: ∀L1,T1,T2. L1 ⊢ T1 ➡ T2 →
 qed-.
 
 (* Basic_1: was by definition: pr2_free *)
-lemma tpr_cpr: ∀T1,T2. ⋆ ⊢ T1 ➡ T2 → ∀L. L ⊢ T1 ➡ T2.
-#T1 #T2 #HT12 #L
-lapply (cpr_lsubr_trans … HT12 L ?) //
+lemma tpr_cpr: ∀G,T1,T2. ⦃G, ⋆⦄ ⊢ T1 ➡ T2 → ∀L. ⦃G, L⦄ ⊢ T1 ➡ T2.
+#G #T1 #T2 #HT12 #L
+lapply (lsubr_cpr_trans … HT12 L ?) //
 qed.
 
-lemma cpqs_cpr: ∀L,T1,T2. L ⊢ T1 ➤* T2 → L ⊢ T1 ➡ T2.
-#L #T1 #T2 #H elim H -L -T1 -T2 // /2 width=1/ /2 width=6/
-qed.
-
-lemma cpss_cpr: ∀L,T1,T2. L ⊢ T1 ▶* T2 → L ⊢ T1 ➡ T2.
-/3 width=1/ qed.
-
 (* Basic_1: includes by definition: pr0_refl *)
-lemma cpr_refl: ∀T,L. L ⊢ T ➡ T.
-/2 width=1/ qed.
+lemma cpr_refl: ∀G,T,L. ⦃G, L⦄ ⊢ T ➡ T.
+#G #T elim T -T // * /2 width=1/
+qed.
 
-lemma cpr_delift: ∀L,K,V,T1,d. ⇩[0, d] L ≡ (K. ⓓV) →
-                  ∃∃T2,T. L ⊢ T1 ➡ T2 & ⇧[d, 1] T ≡ T2.
-#L #K #V #T1 #d #HLK
-elim (cpqs_delift … T1 … HLK) -HLK /3 width=4/
+(* Basic_1: was: pr2_head_1 *)
+lemma cpr_pair_sn: ∀I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 →
+                   ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡ ②{I}V2.T.
+* /2 width=1/ qed.
+
+lemma cpr_delift: ∀G,K,V,T1,L,d. ⇩[0, d] L ≡ (K.ⓓV) →
+                  ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ➡ T2 & ⇧[d, 1] T ≡ T2.
+#G #K #V #T1 elim T1 -T1
+[ * #i #L #d #HLK /2 width=4/
+  elim (lt_or_eq_or_gt i d) #Hid [1,3: /3 width=4/ ]
+  destruct
+  elim (lift_total V 0 (i+1)) #W #HVW
+  elim (lift_split … HVW i i) // /3 width=6/
+| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #d #HLK
+  elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
+  [ elim (IHU1 (L. ⓑ{I}W1) (d+1)) -IHU1 /2 width=1/ -HLK /3 width=9/
+  | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8/
+  ]
+]
 qed-.
 
-lemma cpr_append: l_appendable_sn … cpr.
-#K #T1 #T2 #H elim H -K -T1 -T2 // /2 width=1/ /2 width=3/
-#K #K0 #V1 #V2 #W2 #i #HK0 #_ #HVW2 #IHV12 #L
-lapply (ldrop_fwd_ldrop2_length … HK0) #H
+lemma cpr_append: ∀G. l_appendable_sn … (cpr G).
+#G #K #T1 #T2 #H elim H -G -K -T1 -T2 // /2 width=1/ /2 width=3/
+#G #K #K0 #V1 #V2 #W2 #i #HK0 #_ #HVW2 #IHV12 #L
+lapply (ldrop_fwd_length_lt2 … HK0) #H
 @(cpr_delta … (L@@K0) V1 … HVW2) //
 @(ldrop_O1_append_sn_le … HK0) /2 width=2/ (**) (* /3/ does not work *)
 qed.
 
-lemma cpr_ext_bind: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀V,T1,T2. L.ⓛV ⊢ T1 ➡ T2 →
-                    ∀a,I. L ⊢ ⓑ{a,I}V1. T1 ➡ ⓑ{a,I}V2. T2.
-#L #V1 #V2 #HV12 #V #T1 #T2 #HT12 #a #I
-lapply (cpr_lsubr_trans … HT12 (L.ⓑ{I}V1) ?) -HT12 /2 width=1/
-qed.
-
 (* Basic inversion lemmas ***************************************************)
 
-fact cpr_inv_atom1_aux: ∀L,T1,T2. L ⊢ T1 ➡ T2 → ∀I. T1 = ⓪{I} →
+fact cpr_inv_atom1_aux: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ∀I. T1 = ⓪{I} →
                         T2 = ⓪{I} ∨
-                        ∃∃K,V,V2,i. ⇩[O, i] L ≡ K. ⓓV &
-                                    K ⊢ V ➡ V2 &
-                                    ⇧[O, i + 1] V2 ≡ T2 &
-                                    I = LRef i.
-#L #T1 #T2 * -L -T1 -T2
-[ #I #L #J #H destruct /2 width=1/
-| #L #K #V #V2 #T2 #i #HLK #HV2 #HVT2 #J #H destruct /3 width=8/
-| #a #I #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
-| #I #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
-| #L #V #T1 #T #T2 #_ #_ #J #H destruct
-| #L #V #T1 #T2 #_ #J #H destruct
-| #a #L #V1 #V2 #W #T1 #T2 #_ #_ #J #H destruct
-| #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #J #H destruct
+                        ∃∃K,V,V2,i. ⇩[O, i] L ≡ K. ⓓV & ⦃G, K⦄ ⊢ V ➡ V2 &
+                                    ⇧[O, i + 1] V2 ≡ T2 & I = LRef i.
+#G #L #T1 #T2 * -G -L -T1 -T2
+[ #I #G #L #J #H destruct /2 width=1/
+| #L #G #K #V #V2 #T2 #i #HLK #HV2 #HVT2 #J #H destruct /3 width=8/
+| #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
+| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
+| #G #L #V #T1 #T #T2 #_ #_ #J #H destruct
+| #G #L #V #T1 #T2 #_ #J #H destruct
+| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #J #H destruct
+| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #J #H destruct
 ]
 qed-.
 
-lemma cpr_inv_atom1: ∀I,L,T2. L ⊢ ⓪{I} ➡ T2 →
+lemma cpr_inv_atom1: ∀I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ➡ T2 →
                      T2 = ⓪{I} ∨
-                     ∃∃K,V,V2,i. ⇩[O, i] L ≡ K. ⓓV &
-                                 K ⊢ V ➡ V2 &
-                                 ⇧[O, i + 1] V2 ≡ T2 &
-                                 I = LRef i.
+                     ∃∃K,V,V2,i. ⇩[O, i] L ≡ K. ⓓV & ⦃G, K⦄ ⊢ V ➡ V2 &
+                                 ⇧[O, i + 1] V2 ≡ T2 & I = LRef i.
 /2 width=3 by cpr_inv_atom1_aux/ qed-.
 
 (* Basic_1: includes: pr0_gen_sort pr2_gen_sort *)
-lemma cpr_inv_sort1: ∀L,T2,k. L ⊢ ⋆k ➡ T2 → T2 = ⋆k.
-#L #T2 #k #H
+lemma cpr_inv_sort1: ∀G,L,T2,k. ⦃G, L⦄ ⊢ ⋆k ➡ T2 → T2 = ⋆k.
+#G #L #T2 #k #H
 elim (cpr_inv_atom1 … H) -H //
 * #K #V #V2 #i #_ #_ #_ #H destruct
 qed-.
 
 (* Basic_1: includes: pr0_gen_lref pr2_gen_lref *)
-lemma cpr_inv_lref1: ∀L,T2,i. L ⊢ #i ➡ T2 →
+lemma cpr_inv_lref1: ∀G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡ T2 →
                      T2 = #i ∨
-                     ∃∃K,V,V2. ⇩[O, i] L ≡ K. ⓓV &
-                               K ⊢ V ➡ V2 &
+                     ∃∃K,V,V2. ⇩[O, i] L ≡ K. ⓓV & ⦃G, K⦄ ⊢ V ➡ V2 &
                                ⇧[O, i + 1] V2 ≡ T2.
-#L #T2 #i #H
+#G #L #T2 #i #H
 elim (cpr_inv_atom1 … H) -H /2 width=1/
 * #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=6/
 qed-.
 
-lemma cpr_inv_gref1: ∀L,T2,p. L ⊢ §p ➡ T2 → T2 = §p.
-#L #T2 #p #H
+lemma cpr_inv_gref1: ∀G,L,T2,p. ⦃G, L⦄ ⊢ §p ➡ T2 → T2 = §p.
+#G #L #T2 #p #H
 elim (cpr_inv_atom1 … H) -H //
 * #K #V #V2 #i #_ #_ #_ #H destruct
 qed-.
 
-fact cpr_inv_bind1_aux: ∀L,U1,U2. L ⊢ U1 ➡ U2 →
-                        ∀a,I,V1,T1. U1 = ⓑ{a,I} V1. T1 → (
-                        ∃∃V2,T2. L ⊢ V1 ➡ V2 &
-                                 L. ⓑ{I} V1 ⊢ T1 ➡ T2 &
-                                 U2 = ⓑ{a,I} V2. T2
+fact cpr_inv_bind1_aux: ∀G,L,U1,U2. ⦃G, L⦄ ⊢ U1 ➡ U2 →
+                        ∀a,I,V1,T1. U1 = ⓑ{a,I}V1. T1 → (
+                        ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡ T2 &
+                                 U2 = ⓑ{a,I}V2.T2
                         ) ∨
-                        ∃∃T. L.ⓓV1 ⊢ T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true & I = Abbr.
-#L #U1 #U2 * -L -U1 -U2
-[ #I #L #b #J #W1 #U1 #H destruct
-| #L #K #V #V2 #W2 #i #_ #_ #_ #b #J #W1 #U1 #H destruct
-| #a #I #L #V1 #V2 #T1 #T2 #HV12 #HT12 #b #J #W1 #U1 #H destruct /3 width=5/
-| #I #L #V1 #V2 #T1 #T2 #_ #_ #b #J #W1 #U1 #H destruct
-| #L #V #T1 #T #T2 #HT1 #HT2 #b #J #W1 #U1 #H destruct /3 width=3/
-| #L #V #T1 #T2 #_ #b #J #W1 #U1 #H destruct
-| #a #L #V1 #V2 #W #T1 #T2 #_ #_ #b #J #W1 #U1 #H destruct
-| #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #b #J #W1 #U1 #H destruct
+                        ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡ T & ⇧[0, 1] U2 ≡ T &
+                             a = true & I = Abbr.
+#G #L #U1 #U2 * -L -U1 -U2
+[ #I #G #L #b #J #W1 #U1 #H destruct
+| #L #G #K #V #V2 #W2 #i #_ #_ #_ #b #J #W #U1 #H destruct
+| #a #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #b #J #W #U1 #H destruct /3 width=5/
+| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #b #J #W #U1 #H destruct
+| #G #L #V #T1 #T #T2 #HT1 #HT2 #b #J #W #U1 #H destruct /3 width=3/
+| #G #L #V #T1 #T2 #_ #b #J #W #U1 #H destruct
+| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #b #J #W #U1 #H destruct
+| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #b #J #W #U1 #H destruct
 ]
 qed-.
 
-lemma cpr_inv_bind1: ∀a,I,L,V1,T1,U2. L ⊢ ⓑ{a,I} V1. T1 ➡ U2 → (
-                     ∃∃V2,T2. L ⊢ V1 ➡ V2 &
-                              L. ⓑ{I} V1 ⊢ T1 ➡ T2 &
-                              U2 = ⓑ{a,I} V2. T2
+lemma cpr_inv_bind1: ∀a,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡ U2 → (
+                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡ T2 &
+                              U2 = ⓑ{a,I}V2.T2
                      ) ∨
-                     ∃∃T. L.ⓓV1 ⊢ T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true & I = Abbr.
+                     ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡ T & ⇧[0, 1] U2 ≡ T &
+                          a = true & I = Abbr.
 /2 width=3 by cpr_inv_bind1_aux/ qed-.
 
 (* Basic_1: includes: pr0_gen_abbr pr2_gen_abbr *)
-lemma cpr_inv_abbr1: ∀a,L,V1,T1,U2. L ⊢ ⓓ{a} V1. T1 ➡ U2 → (
-                     ∃∃V2,T2. L ⊢ V1 ➡ V2 &
-                              L. ⓓ V1 ⊢ T1 ➡ T2 &
-                              U2 = ⓓ{a} V2. T2
+lemma cpr_inv_abbr1: ∀a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡ U2 → (
+                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L. ⓓV1⦄ ⊢ T1 ➡ T2 &
+                              U2 = ⓓ{a}V2.T2
                      ) ∨
-                     ∃∃T. L.ⓓV1 ⊢ T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true.
-#a #L #V1 #T1 #U2 #H
+                     ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true.
+#a #G #L #V1 #T1 #U2 #H
 elim (cpr_inv_bind1 … H) -H * /3 width=3/ /3 width=5/
 qed-.
 
 (* Basic_1: includes: pr0_gen_abst pr2_gen_abst *)
-lemma cpr_inv_abst1: ∀a,L,V1,T1,U2. L ⊢ ⓛ{a} V1. T1 ➡ U2 →
-                     ∃∃V2,T2. L ⊢ V1 ➡ V2 & L. ⓛ V1 ⊢ T1 ➡ T2 &
-                              U2 = ⓛ{a} V2. T2.
-#a #L #V1 #T1 #U2 #H
+lemma cpr_inv_abst1: ∀a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 ➡ U2 →
+                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡ T2 &
+                              U2 = ⓛ{a}V2.T2.
+#a #G #L #V1 #T1 #U2 #H
 elim (cpr_inv_bind1 … H) -H *
 [ /3 width=5/
 | #T #_ #_ #_ #H destruct
 ]
 qed-.
 
-fact cpr_inv_flat1_aux: ∀L,U,U2. L ⊢ U ➡ U2 →
-                        ∀I,V1,U1. U = ⓕ{I} V1. U1 →
-                        ∨∨ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ U1 ➡ T2 &
+fact cpr_inv_flat1_aux: ∀G,L,U,U2. ⦃G, L⦄ ⊢ U ➡ U2 →
+                        ∀I,V1,U1. U = ⓕ{I}V1.U1 →
+                        ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ U1 ➡ T2 &
                                     U2 = ⓕ{I} V2. T2
-                         | (L ⊢ U1 ➡ U2 ∧ I = Cast)
-                         | ∃∃a,V2,W,T1,T2. L ⊢ V1 ➡ V2 & L.ⓛW ⊢ T1 ➡ T2 &
-                                           U1 = ⓛ{a}W. T1 &
-                                           U2 = ⓓ{a}V2. T2 & I = Appl
-                         | ∃∃a,V,V2,W1,W2,T1,T2. L ⊢ V1 ➡ V & ⇧[0,1] V ≡ V2 &
-                                                 L ⊢ W1 ➡ W2 & L.ⓓW1 ⊢ T1 ➡ T2 &
-                                                 U1 = ⓓ{a}W1. T1 &
-                                                 U2 = ⓓ{a}W2. ⓐV2. T2 & I = Appl.
-#L #U #U2 * -L -U -U2
-[ #I #L #J #W1 #U1 #H destruct
-| #L #K #V #V2 #W2 #i #_ #_ #_ #J #W1 #U1 #H destruct
-| #a #I #L #V1 #V2 #T1 #T2 #_ #_ #J #W1 #U1 #H destruct
-| #I #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W1 #U1 #H destruct /3 width=5/
-| #L #V #T1 #T #T2 #_ #_ #J #W1 #U1 #H destruct
-| #L #V #T1 #T2 #HT12 #J #W1 #U1 #H destruct /3 width=1/
-| #a #L #V1 #V2 #W #T1 #T2 #HV12 #HT12 #J #W1 #U1 #H destruct /3 width=9/
-| #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #J #W1 #U1 #H destruct /3 width=13/
+                         | (⦃G, L⦄ ⊢ U1 ➡ U2 ∧ I = Cast)
+                         | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ W1 ➡ W2 &
+                                               ⦃G, L.ⓛW1⦄ ⊢ T1 ➡ T2 & U1 = ⓛ{a}W1.T1 &
+                                               U2 = ⓓ{a}ⓝW2.V2.T2 & I = Appl
+                         | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V & ⇧[0,1] V ≡ V2 &
+                                                 ⦃G, L⦄ ⊢ W1 ➡ W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡ T2 &
+                                                 U1 = ⓓ{a}W1.T1 &
+                                                 U2 = ⓓ{a}W2.ⓐV2.T2 & I = Appl.
+#G #L #U #U2 * -L -U -U2
+[ #I #G #L #J #W1 #U1 #H destruct
+| #G #L #K #V #V2 #W2 #i #_ #_ #_ #J #W #U1 #H destruct
+| #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #W #U1 #H destruct
+| #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W #U1 #H destruct /3 width=5/
+| #G #L #V #T1 #T #T2 #_ #_ #J #W #U1 #H destruct
+| #G #L #V #T1 #T2 #HT12 #J #W #U1 #H destruct /3 width=1/
+| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12 #J #W #U1 #H destruct /3 width=11/
+| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #J #W #U1 #H destruct /3 width=13/
 ]
 qed-.
 
-lemma cpr_inv_flat1: ∀I,L,V1,U1,U2. L ⊢ ⓕ{I} V1. U1 ➡ U2 →
-                     ∨∨ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ U1 ➡ T2 &
-                                 U2 = ⓕ{I} V2. T2
-                      | (L ⊢ U1 ➡ U2 ∧ I = Cast)
-                      | ∃∃a,V2,W,T1,T2. L ⊢ V1 ➡ V2 & L.ⓛW ⊢ T1 ➡ T2 &
-                                        U1 = ⓛ{a}W. T1 &
-                                        U2 = ⓓ{a}V2. T2 & I = Appl
-                      | ∃∃a,V,V2,W1,W2,T1,T2. L ⊢ V1 ➡ V & ⇧[0,1] V ≡ V2 &
-                                              L ⊢ W1 ➡ W2 & L.ⓓW1 ⊢ T1 ➡ T2 &
-                                              U1 = ⓓ{a}W1. T1 &
-                                              U2 = ⓓ{a}W2. ⓐV2. T2 & I = Appl.
+lemma cpr_inv_flat1: ∀I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ➡ U2 →
+                     ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ U1 ➡ T2 &
+                                 U2 = ⓕ{I}V2.T2
+                      | (⦃G, L⦄ ⊢ U1 ➡ U2 ∧ I = Cast)
+                      | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ W1 ➡ W2 &
+                                            ⦃G, L.ⓛW1⦄ ⊢ T1 ➡ T2 & U1 = ⓛ{a}W1.T1 &
+                                            U2 = ⓓ{a}ⓝW2.V2.T2 & I = Appl
+                      | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V & ⇧[0,1] V ≡ V2 &
+                                              ⦃G, L⦄ ⊢ W1 ➡ W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡ T2 &
+                                              U1 = ⓓ{a}W1.T1 &
+                                              U2 = ⓓ{a}W2.ⓐV2.T2 & I = Appl.
 /2 width=3 by cpr_inv_flat1_aux/ qed-.
 
 (* Basic_1: includes: pr0_gen_appl pr2_gen_appl *)
-lemma cpr_inv_appl1: ∀L,V1,U1,U2. L ⊢ ⓐ V1. U1 ➡ U2 →
-                     ∨∨ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ U1 ➡ T2 &
-                                 U2 = ⓐ V2. T2
-                      | ∃∃a,V2,W,T1,T2. L ⊢ V1 ➡ V2 & L.ⓛW ⊢ T1 ➡ T2 &
-                                        U1 = ⓛ{a}W. T1 & U2 = ⓓ{a}V2. T2
-                      | ∃∃a,V,V2,W1,W2,T1,T2. L ⊢ V1 ➡ V & ⇧[0,1] V ≡ V2 &
-                                              L ⊢ W1 ➡ W2 & L.ⓓW1 ⊢ T1 ➡ T2 &
-                                              U1 = ⓓ{a}W1. T1 & U2 = ⓓ{a}W2. ⓐV2. T2.
-#L #V1 #U1 #U2 #H elim (cpr_inv_flat1 … H) -H *
+lemma cpr_inv_appl1: ∀G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐV1.U1 ➡ U2 →
+                     ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ U1 ➡ T2 &
+                                 U2 = ⓐV2.T2
+                      | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ W1 ➡ W2 &
+                                            ⦃G, L.ⓛW1⦄ ⊢ T1 ➡ T2 &
+                                            U1 = ⓛ{a}W1.T1 & U2 = ⓓ{a}ⓝW2.V2.T2
+                      | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V & ⇧[0,1] V ≡ V2 &
+                                              ⦃G, L⦄ ⊢ W1 ➡ W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡ T2 &
+                                              U1 = ⓓ{a}W1.T1 & U2 = ⓓ{a}W2.ⓐV2.T2.
+#G #L #V1 #U1 #U2 #H elim (cpr_inv_flat1 … H) -H *
 [ /3 width=5/
 | #_ #H destruct
-| /3 width=9/
+| /3 width=11/
 | /3 width=13/
 ]
 qed-.
 
 (* Note: the main property of simple terms *)
-lemma cpr_inv_appl1_simple: ∀L,V1,T1,U. L ⊢ ⓐV1. T1 ➡ U → 𝐒⦃T1⦄ →
-                            ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ T1 ➡ T2 &
+lemma cpr_inv_appl1_simple: ∀G,L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1. T1 ➡ U → 𝐒⦃T1⦄ →
+                            ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ T1 ➡ T2 &
                                      U = ⓐV2. T2.
-#L #V1 #T1 #U #H #HT1
+#G #L #V1 #T1 #U #H #HT1
 elim (cpr_inv_appl1 … H) -H *
 [ /2 width=5/
-| #a #V2 #W #U1 #U2 #_ #_ #H #_ destruct
+| #a #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #H #_ destruct
   elim (simple_inv_bind … HT1)
 | #a #V #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct
   elim (simple_inv_bind … HT1)
@@ -268,50 +266,33 @@ elim (cpr_inv_appl1 … H) -H *
 qed-.
 
 (* Basic_1: includes: pr0_gen_cast pr2_gen_cast *)
-lemma cpr_inv_cast1: ∀L,V1,U1,U2. L ⊢ ⓝ V1. U1 ➡ U2 → (
-                     ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ U1 ➡ T2 &
+lemma cpr_inv_cast1: ∀G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝ V1. U1 ➡ U2 → (
+                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ U1 ➡ T2 &
                               U2 = ⓝ V2. T2
-                     ) ∨
-                     L ⊢ U1 ➡ U2.
-#L #V1 #U1 #U2 #H elim (cpr_inv_flat1 … H) -H *
+                     ) ∨ ⦃G, L⦄ ⊢ U1 ➡ U2.
+#G #L #V1 #U1 #U2 #H elim (cpr_inv_flat1 … H) -H *
 [ /3 width=5/
 | /2 width=1/
-| #a #V2 #W #T1 #T2 #_ #_ #_ #_ #H destruct
+| #a #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #H destruct
 | #a #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #H destruct
 ]
 qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma cpr_fwd_abst1: ∀a,L,V1,T1,U2. L ⊢ ⓛ{a}V1.T1 ➡ U2 → ∀I,W.
-                     ∃∃V2,T2. L ⊢ V1 ➡ V2 & L. ⓑ{I} W ⊢ T1 ➡ T2 &
-                              U2 = ⓛ{a} V2. T2.
-#a #L #V1 #T1 #U2 #H #I #W
-elim (cpr_inv_abst1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
-lapply (cpr_lsubr_trans … HT12 (L.ⓑ{I}W) ?) -HT12 /2 width=1/ /2 width=5/
-qed-.
-
-
-lemma cpr_fwd_ext_abst1: ∀a,L,V1,T1,U2. L ⊢ ⓛ{a}V1.T1 ➡ U2 → ∀b,I,W.
-                         ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ ⓑ{b,I}W.T1 ➡ ⓑ{b,I}W.T2 &
-                                  U2 = ⓛ{a}V2.T2.
-#a #L #V1 #T1 #U2 #H #b #I #W
-elim (cpr_fwd_abst1 … H I W) -H /3 width=5/
-qed-.
-
-lemma cpr_fwd_bind1_minus: ∀I,L,V1,T1,T. L ⊢ -ⓑ{I}V1.T1 ➡ T → ∀b.
-                           ∃∃V2,T2. L ⊢ ⓑ{b,I}V1.T1 ➡ ⓑ{b,I}V2.T2 &
+lemma cpr_fwd_bind1_minus: ∀I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ➡ T → ∀b.
+                           ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{b,I}V1.T1 ➡ ⓑ{b,I}V2.T2 &
                                     T = -ⓑ{I}V2.T2.
-#I #L #V1 #T1 #T #H #b
+#I #G #L #V1 #T1 #T #H #b
 elim (cpr_inv_bind1 … H) -H *
 [ #V2 #T2 #HV12 #HT12 #H destruct /3 width=4/
-| #T2 #_ #_ #H destruct 
+| #T2 #_ #_ #H destruct
 ]
 qed-.
 
-lemma cpr_fwd_shift1: ∀L1,L,T1,T. L ⊢ L1 @@ T1 ➡ T →
+lemma cpr_fwd_shift1: ∀G,L1,L,T1,T. ⦃G, L⦄ ⊢ L1 @@ T1 ➡ T →
                       ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
-#L1 @(lenv_ind_dx … L1) -L1 normalize
+#G #L1 @(lenv_ind_dx … L1) -L1 normalize
 [ #L #T1 #T #HT1
   @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
 | #I #L1 #V1 #IH #L #T1 #X
@@ -326,13 +307,12 @@ lemma cpr_fwd_shift1: ∀L1,L,T1,T. L ⊢ L1 @@ T1 ➡ T →
 ]
 qed-.
 
-(* Basic_1: removed theorems 12:
+(* Basic_1: removed theorems 11:
             pr0_subst0_back pr0_subst0_fwd pr0_subst0
             pr2_head_2 pr2_cflat clear_pr2_trans
             pr2_gen_csort pr2_gen_cflat pr2_gen_cbind
-            pr2_subst1
             pr2_gen_ctail pr2_ctail
-*)   
+*)
 (* Basic_1: removed local theorems 4:
             pr0_delta_tau pr0_cong_delta
             pr2_free_free pr2_free_delta